Applied Statistics and Probability for Engineers

8-1

8-2.6 Bootstrap Confidence Intervals (CD Only)

In Section 7-2.5 we showed how a technique called the bootstrapcould be used to estimate

the standard error where is an estimate of a parameter . We can also use the bootstrap

to find a confidence interval on the parameter . To illustrate, consider the case where is the

mean of a normal distribution with known. Now the estimator of is Also notice that

is the 100(1 /2) percentile of the distribution of , and is

the 100(2) percentile of this distribution. Therefore, we can write the probability statement

associated with the 100(1 )% confidence interval as

or

This last probability statement implies that the lower and upper 100(1 )% confidence lim-

its for are

We may generalize this to an arbitrary parameter . The 100(1)% confidence limits

for are

Unfortunately, the percentiles of may not be as easy to find as in the case of the normal

distribution mean. However, they could be estimated from bootstrap samples.Suppose we

find Bbootstrap samples and calculate , , p, and and then calculate

, p,. The required percentiles can be obtained directly from the differences.

For example, if B200 and a 95% confidence interval on is desired, the fifth smallest and

fifth largest of the differences are the estimates of the necessary percentiles.

We will illustrate this procedure using the situation first described in Example 7-3,

involving the parameter of an exponential distribution. Following that example, a random

sample of n8 engine controller modules were tested to failure, and the estimate of 

obtained was 0.0462, where is a maximum likelihood estimator. We used 200

bootstrap samples to obtain an estimate of the standard error for.

Figure S8-1(a) is a histogram of the 200 bootstrap estimates , i 1, 2, p, 200. Notice

that the histogram is not symmetrical and is skewed to the right, indicating thatthe sam-

pling distribution of also has this same shape. We subtracted the sample average of these

bootstrap estimates 0.5013 from each. The histogram of the differences , i

1, 2, p, 200, is shown in Figure S8-1(b). Suppose we wish to find a 90% confidence inter-

val for . Now the fifth percentile of the bootstrap samples is0.0228 and the ninety-

fifth percentile is 0.03135. Therefore the lower and upper 90% bootstrap confidence limits are

Uˆ5 percentile of ˆ*i* 0.0462 1 0.0228 2 0.0690

Lˆ95 percentile of ˆ*i* 0.04620.031350.0149

ˆ*i*

* ˆi* ˆi**

ˆ

ˆ*i

ˆ

ˆ ˆ (^1) X
ˆi
ˆ 2  ˆB
ˆ 1 ˆ 2 ˆB  ˆ 1 ,
ˆ
Uˆ 1001  22 percentile of ˆ
Lˆ 10011  22 percentile of ˆ
UX 100 1  22 percentile of XX z 2  1 n
LX 100 11  22 percentile of XXz 2  1 n
P 1 X 10011  22 percentile

X 1001  22 percentile 2  1 
P 11001  22 percentile X
10011  22 percentile 2  1 
z 2  1 n X z 2  1 n
X.
ˆ, ˆ
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